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A304974
Number of achiral color patterns (set partitions) for a row or loop of length n using exactly 4 colors (sets).
9
0, 0, 0, 0, 1, 2, 9, 16, 53, 90, 265, 440, 1221, 2002, 5369, 8736, 22933, 37130, 96105, 155080, 397541, 640002, 1629529, 2619056, 6636213, 10653370, 26899145, 43144920, 108659461, 174174002, 437826489, 701478976, 1760871893, 2820264810, 7072185385, 11324105960, 28374834981, 45425564002, 113757620249
OFFSET
0,6
COMMENTS
Two color patterns are equivalent if we permute the colors. Achiral color patterns must be equivalent if we reverse the order of the pattern.
FORMULA
a(n) = [n==0 mod 2] * (S2(n/2+2, 4) - S2(n/2+1, 4) - 2*S2(n/2, 4)) + [n==1 mod 2] * (2*S2((n+3)/2, 4) - 4*S2((n+1)/2, 4)) where S2(n,k) is the Stirling subset number A008277(n,k).
G.f.: x^4 * (1+x)^2 * (1-2x^2) / Product_{k=1..4} (1 - k*x^2).
a(n) = A304972(n,4).
a(2m-1) = A140735(m,4).
a(2m) = A293181(m,4).
EXAMPLE
For a(6) = 9, the row color patterns are AABCDD, ABACDC, ABBCCD, ABCADC, ABCBCD, ABCCBD, ABCCDA, ABCDAB, and ABCBCD. The loop color patterns are AAABCD, AABBCD, AABCCD, AABCDB, ABABCD, ABACAD, ABACBD, ABACDC, and ABCADC.
MATHEMATICA
Table[If[EvenQ[n], StirlingS2[n/2 + 2, 4] - StirlingS2[n/2 + 1, 4] - 2 StirlingS2[n/2, 4], 2 StirlingS2[(n + 3)/2, 4] - 4 StirlingS2[(n + 1)/2, 4]], {n, 0, 40}]
Join[{0}, LinearRecurrence[{1, 7, -7, -12, 12}, {0, 0, 0, 1, 2}, 40]] (* Robert A. Russell, Oct 14 2018 *)
PROG
(PARI) m=40; v=concat([0, 0, 0, 1, 2], vector(m-5)); for(n=6, m, v[n] = v[n-1] +7*v[n-2] -7*v[n-3] -12*v[n-4] +12*v[n-5]); concat([0], v) \\ G. C. Greubel, Oct 17 2018
(Magma) I:=[0, 0, 0, 1, 2]; [0] cat [n le 5 select I[n] else Self(n-1) +7*Self(n-2) -7*Self(n-3) -12*Self(n-4) +12*Self(n-5): n in [1..40]]; // G. C. Greubel, Oct 17 2018
CROSSREFS
Fourth column of A304972.
Fourth column of A140735 for odd n.
Fourth column of A293181 for even n.
Coefficients that determine the first formula and generating function are row 4 of A305008.
Sequence in context: A237282 A178440 A097965 * A075645 A185252 A055259
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, May 22 2018
STATUS
approved